Expansion of the inverse of positive‐definite Toeplitz operators over polytopes

Article type: Research Article

Authors: Kateb, Djalil | Seghier, Abdellatif

Affiliations: U.T.C. Compiegne, D.M.A. Royallieu, 60200 Compiegne, France | Université de Paris‐Sud, Batiment 405, 91000 Orsay, France

Abstract: Let \tilde\varLambda be a polytope in \mathbb{R}^d, \varLambda =\tilde\varLambda \cap\mathbb{Z}^{d} be its trace on the group \mathbb{Z}^{d} and let T_{\varLambda }(f) be a Toeplitz operator, with a positive symbol f\in L^{\infty}(\mathbb{T}^{d}), defined on the Hardy space H^2(\varLambda ). With some additional assumptions on f, a purely algebraic inverse formula of T_{\varLambda }(f) is given. The geometric properties of the operators are highlighted when \varLambda is inflated to get \varLambda _\lambda =\lambda \varLambda with \lambda=2^{m} chosen sufficiently large. A localization result for the inverse operator is obtained. When we are sufficiently close to a (d-k)‐dimensional face of the inflated polytope, the localized inversion formula is reduced to those operators reflecting this proximity. Subsequently an analogue of the strong Szegő limit theorem is stated; a d+1 order asymptotics formula of the trace of the inverse is given that connects each operator of the sum with some geometric measures of our polytope (volume, area,\,\ldots). Furthermore, the results of Thorsen and Doktorski are retrieved with the help of the Ehrart lattice enumerator formula.

Journal: Asymptotic Analysis, vol. 22, no. 3-4, pp. 205-234, 2000